Euclidean Geometry is the geometry of the flat plane.
Euclid's Elements is a systematic study of the theorems and propositions that can be proven from the axioms.
For centuries it was thought that Euclidean Geometry was somehow "The" geometry, and that the axioms were self-evidently true in our world.
Nikolai Lobachevsky (1792 - 1856) and JŠnos Bolyai (1802 - 1860) proved independently that there were models of the first four axioms that did not satisfy the fifth, thus showing that the fifth postulate cannot be proven from the other four.
By using alternative versions of the fifth postulate we obtain so-called Non-Euclidean Geometry.
Some of this material is duplicated on the page about Axioms, from which it could perhaps be removed. ALternatively, an overview page on Geometry could be written.